When one calculate the amplitude for a particle to propagate between two points, the results seems to violate causality. One book that makes some comments about this is Peskin & Schroeder, chapter 2, page 10. They use a square-root Hamiltonian. Even if we use the local Klein-Gordon equation and we start with say a delta function, then the wave function will spread superluminally.
In the case of a relativistic string we can follow Polchinski's book for example and we get the string spectrum (page 23) but there is nothing about causality or superluminal propagation.
So again if we start with a localized wave packet for the string, will this wave packet spread superluminally? If yes, that would mean that string theory is inconsistent?
I am currently a beginer in string theory and qft but I think this should be an important question addressed at the begining. Regarding the calculation for the amplitude for a string to propagate between two spacetime points I have not found any other than Emil Martinec, arXiv:hep-th/9304037, but it seems that he is using String Field Theory. But that paper makes me think that: as in particle theory we have to ensure causality going from first quantization to QFT, we need to address the analogous question in string theory (and perhaps go unavoidably to SFT?).
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